Backlinks have no negative influence they have value or they haven't. The links on your website how ever do matter if you link to a spam or what so ever website you can loose some ranking.
If Google bans you, you do something wrong and most likely other will ban you too. How ever it is interesting to use some combination with Adsense so earnings are maximized.
If you moved your content then yes it will help just helping old URLs towards a random site won't help, maybe in the beginning but for long term it won't.
First of all you should not care so much about pagerank.
The differences in the Search-engine and Webmaster can be several reasons one could be the downtime you mentioned. If I were you I would not worry to much, time is a important SEO factor.
In a nutshell: Web site owners use the /robots.txt file to give instructions about their site to web robots; this is called The Robots Exclusion Protocol. http://www.robotstxt.org/robotstxt.html
Well if you ask me Joomla is not a good CMS for SEO purposes you need to change and hack to much while other systems as Wordpress or Drupal offer a lot more out of the box.
I earn about 30-50 dollars a month must say that it doesn't grow much anymore while my traffic does. I think it is because Google Ads are getting to known by the audience anyone experience with a other service?